Abstract
In this paper we consider pricing problems of the geometric average Asian options under a non-Gaussian model, in which the underlying stock price is driven by a process based on non-extensive statistical mechanics. The model can describe the peak and fat tail characteristics of returns. Thus, the description of underlying asset price and the pricing of options are more accurate. Moreover, using the martingale method, we obtain closed form solutions for geometric average Asian options. Furthermore, the numerical analysis shows that the model can avoid underestimating risks relative to the Black-Scholes model.
Highlights
It is very important for investors to accurately describe the law of asset price movement, which is the foundation of the pricing and risk management of derivatives
It is well known that the hypothesis that the asset price changes follow Brownian motion implies that the price changes are independent and the distributions of log-returns are normal, but a number of studies have shown that empirical distributions of returns are not normal distributions
Queirós and Moyano [11], and Biró and Rosenfeld [12] found that the empirical distribution of returns of the Dow Jones Industrial Average can be well fitted by a Tsallis distribution
Summary
It is very important for investors to accurately describe the law of asset price movement, which is the foundation of the pricing and risk management of derivatives. The use of geometric Brownian motion to describe the underlying stock price is inaccurate This usually makes the price of options higher than the actual value and causes investors to underestimate the risk. Chung et al [27] employed a mean reversion and jump process to describe the movement of underlying stock price instead of the geometric Brownian motion and studied the pricing of arithmetic average Asian options. In order to describe the movement of underlying stock price more accurately, we consider an asset price model with a Tsallis distribution of index q in the framework of non-extensive statistical mechanics. We study the pricing of geometric average Asian options and obtain closed form solutions by the use of the martingale method.
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